Existence of solutions with prescribed norm for semilinear elliptic equations. (English) Zbl 0877.35091

The nonlinear eigenvalue problem \[ -\Delta u(x) - g(u(x)) = \lambda u(x),\quad\lambda \in \mathbb{R},\;x \in \mathbb{R}^N \] is studied. It is supposed that \(g:\mathbb{R} \to \mathbb{R}\) is continuous, odd, and such that \[ \alpha \int_0^s g(t) dt \leq g(s)s \leq \beta \int_0^s g(t) dt \] with some \((2N+4)/N<\alpha\leq\beta< (2N)/(N-2)\) if \(N\geq 3\) or \((2N+4)/N<\alpha\leq\beta\) if \(n=1,2\). A certain mild additional assumption is added in the case \(N=1\). It is proved that for any \(c>0\) there exists a weak solution \(u_c \in H^1(\mathbb{R}^N),\;\lambda_c \in \mathbb{R}\) satisfying \(|u|_{L^2}= c,\;\lambda_c <0\). The proof is based on a minimax approach used for the corresponding functional. Further, the dependence of \(|\nabla u_c|_{L^2}\) and \(\lambda_c\) on \(c\) is described. Particularly, it follows \(|\nabla u_c|_{L^2} \to +\infty,\;\lambda_c \to -\infty\) as \(c \to 0\) and \(|\nabla u_c|_{L^2} \to 0,\;\lambda_c \to 0\) as \(c \to +\infty\).
Reviewer: M.Kučera (Praha)


35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
Full Text: DOI


[1] Ambrosetti, A.; Bertotti, M. L., Homoclinics for second order convervative systems. Partial Differential Equations and Related Subjects, (M., Miranda, Pitman Reseach Note in Math. Ser. (1992)) · Zbl 0804.34046
[2] Ambrosetti, A.; Struwe, M., Existence of steady vortex rings in an ideal fluid, Arch. Rat. Mech. Anal., 108, 97-109 (1989) · Zbl 0694.76012
[3] Benci, V.; Cerami, G., Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rat. Mech. Anal., 99, 283-300 (1987) · Zbl 0635.35036
[4] Berestycki, H.; Lions, P. L., Nonlinear scalar field equations I, Arch. Rat. Mech. Anal., 82, 313-346 (1983)
[5] Berestycki, H.; Lions, P. L., Nonlinear scalar field equations II, Arch. Rat. Mech. Anal., 82, 347-376 (1983)
[6] Brezis, H.; Kato, T., Remarks on the Schrodinger operator with singular complex potentials, J. Mat. Pures Appli., 58, 137-151 (1979) · Zbl 0408.35025
[7] Buffoni, B., Un problème variationnel fortement indéfini sans compacité, (Ph.D. thesis (1992), EPFL: EPFL New York)
[9] Buffoni, B.; Jeanjean, L., Minimax characterization of solutions for a semilinear elliptic equation, Ann. Inst. H. Poincaré, Anal. non-lin., 10, 4, 377 (1993) · Zbl 0828.35013
[10] Jeanjean, L., Approche minimax des solutions d’une équation semi-linéaire elliptique en l’absence de compacité, (Ph.D. thesis (1992), EPFL)
[11] Lions, J. L., Problèmes aux limites dans les équations aux dérivés partielles (1962), Presses de l’univ.
[12] Lions, P. L., The concentration-compactness principle in the calculus of variations, Part 1, Ann. Inst. H. Poincaré, Anal, non-lin., 1, 109-145 (1984) · Zbl 0541.49009
[13] Lions, P. L., The concentration-compactness principle in the calculus of variations, Part 2, Ann. Inst. H. Poincaré, Anal, non-lin., 1, 223-283 (1984) · Zbl 0704.49004
[14] Krasnoselskii, M. A., Topological Methods in the Theory of Nonlinear Integral Operators (1964), Pergamon Press: Pergamon Press de Montréal
[15] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, Appli. Math. Sci., 74 (1989) · Zbl 0676.58017
[16] Pohozaev, S., Eigenfunctions of the equations Δu + λƒ (u) = 0, Soviet Math. Dokl., 6, 1408-1411 (1965) · Zbl 0141.30202
[17] Rabinowitz, P. H., Minimax methods in critical point theory with applications to differential equations, (CBMS Conference Lectures (1986), A.M.S.: A.M.S. London) · Zbl 0152.10003
[18] Strauss, W. A., Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55, 149-162 (1977) · Zbl 0356.35028
[19] Struwe, M., (Variational Methods. (1990), Springer: Springer Providence)
[20] Stuart, C. A., Bifurcation for variational problems when the linearization has no eigenvalues, J. Funct. Anal., 38, 169-187 (1980) · Zbl 0458.47048
[21] Stuart, C. A., Bifurcation from the continuous spectrum in \(L^2\)-theory of elliptic equations on \(\textbf{R}^n \), (RecentMethods in Nonlinear Analysis and Applications (1981), Liguori: Liguori Berlin)
[22] Stuart, C. A., Bifurcation in \(L^p (\textbf{R}^N )\) for a semilinear elliptic equation, (Proc. London Math. Soc., 57 (1988)), 511-541, 3 · Zbl 0673.35005
[23] Stuart, C. A., Bifurcation from the essential spectrum for some non-compact nonlinearities, Math. Appl. Sci., 11, 525-542 (1989) · Zbl 0678.58013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.